Finally, let us emphasise that the action of all these shift operators is general and not in any way tied to the simplex representation: any solution of the 4-point conformal Ward identities is mapped to an appropriately shifted solution.Up to a technical caveat (common to all shift operators) that where divergences occur, one must work in a suitable dimensional regularisation scheme. In some cases the shift operator then only yields the leading divergences of the shifted correlator, see the discussion in [1]}.
We have confirmed this explicitly by computing all the relevant intertwining relations between the shift operators in this section and the conformal Ward identities, whose form in Mandelstam variables can be found in e.g., [1]}, [3]}. Thus, for example,
\(\mathcal {K}(\Delta _1+1,\Delta _2+1,\Delta _3,\Delta _4,d+2)S_{12}^{++} = S_{12}^{++}\mathcal {K}(\Delta _1,\Delta _2,\Delta _3,\Delta _4,d)\) 
